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Keywords:

  • spray concentration profile;
  • ice accretion;
  • spume;
  • spray generation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

We compile measurements of sea spray droplet concentrations near the ocean surface for wind speeds from 0 to 28.8 m s−1. We plot each concentration distribution with the Andreas/Fairall spray generation function for that wind speed to display the production velocity distribution that is required for them to be compatible. The comparison shows that the equilibrium assumption is not consistent with this spray generation function. As an alternative, we try a spray concentration function from the literature and find that it represents the data well for moderate to high wind speeds. Using the compiled data, we then extend this concentration function to the very high wind speeds that generate spume droplets. This function has a stronger dependence on wind speed and a longer tail than the concentration function for jet and film droplets produced by bursting bubbles in whitecaps. To validate these concentration functions, we use a simple ice accretion model with weather data and icing observations from two offshore platforms. The results show that moderate-to-high wind speeds that generate film and jet droplets result in small ice accumulations. However, larger spume droplets created at very high wind speeds produce high icing rates. The spray generation function that is consistent with the equilibrium assumption has a median volume droplet radius characteristic of jet droplets for moderate and high wind speeds and a radius that is characteristic of spume droplets at very high speeds. Copyright © 2011 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

Sea spray droplets are carried by the wind and collide with objects in their path. When the air temperature is below 0°C, spray droplets may accrete as ice on offshore structures. On vessels, spray impacting the superstructure is created primarily by the vessel slamming into waves and swell as it powers through the water. However, many offshore structures, including semi-submersible oil exploration and production platforms, are fixed and have little area at the waterline. For those structures, sea spray impacting the superstructure comes from wind waves.

In this article we examine the relationship between measured sea spray droplet concentrations and the Andreas (2002) spray generation function as a means to estimate the concentration profile. We extend the Lewis and Schwartz (2004) sea spray concentration function to higher wind speeds using concentration data that we compile from the literature. We then simulate sea spray icing based on these droplet concentration functions using weather data from two oil exploration platforms. The icing observations from the platforms provide a check of the concentration functions. Finally, we discuss the spray generation functions that are derived from the concentration functions with the equilibrium assumption.

In section 2 we compile measurements of spray concentration distributions dC/dr from many experiments over the ocean (Monahan, 1968; Preobrazhenskii, 1973; Monahan et al., 1983; de Leeuw, 1986a,b, 1987, 1990; Taylor and Wu, 1992; Smith et al., 1993) and graphically compare them with the Andreas (2002) sea spray generation function dF/dr. We discuss the discrepancy between the generation function and concentration distributions under the assumption that equilibrium requires that they be related by the gravitational fall velocity vg of the droplets.

In section 3 we compare an empirical concentration function for film and jet droplets generated by the bursting of bubbles in whitecaps (Lewis and Schwartz, 2004) with the measured concentrations compiled in section 2. That function applies to wind speeds up to 20 m s−1. We then propose different parameters for a very high wind speed version of the function, to be consistent with concentration distributions measured at a limited range of very high winds.

In section 4 we compare the Fairall et al. (2009) spray concentration profile function with concentration ratios from de Leeuw's measurements (1986a,b, 1987, 1990). The source height for the profiles is specified at 1 m for droplets generated from whitecaps and at 0.5H1/3 for spume droplets, where H1/3 is the significant wave height.

In section 5 we present a simple sea spray icing model based on the concentration functions from section 3, the spray profile from section 4 and collision efficiencies from Finstad et al. (1988). We provide the available observations of weather and sea spray icing on oil exploration platforms and compare the icing observations to our ice accretion rates and thicknesses simulated from the weather data. The section concludes with a discussion of the implications for spray generation.

Our conclusions are presented in section 6 along with suggestions for further work.

2. Sea spray generation and concentration distributions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

2.1. Sea spray generation

Many authors have contributed to our current understanding of the flux of sea spray from the ocean surface. Andreas (2002) reviewed the published spray generation functions, whose values ranged over many orders of magnitude at any given droplet radius, and from them chose the more reliable functions. The current Andreas spray generation function spans droplet diameters from 0.5 to 500 µm, including film, jet and spume droplets, and is valid for wind speeds up to 25 m s−1. It is based on the function from Fairall et al. (1994) and is extended to smaller droplet sizes using the bubbles-only function of Monahan et al. (1986). It relies on the Monahan and O'Muircheartaigh (1980) fractional whitecap coverage function for its wind speed dependence.

In the Andreas formulation, the spray generation function dF/dr80 is calculated from U10, the wind speed at 10 m, as a function of r80, the droplet radius at equilibrium at 80% relative humidity. Then dF/dr0, the spray generation function in terms of r0, the radius at formation at equilibrium with the seawater, is calculated from dF/dr80. Function dF/dr0 gives the number of droplets of radius r0 produced per square metre of sea surface per second per micrometre increment in droplet radius (m−2s−1 µm−1). It varies slightly with salinity because the equilibrium humidity at formation depends on the salinity of the seawater. The variation of dF/dr0 with radius for salinity S = 30‰ and U10 = 12 m s−1 and for winds half and twice as high is shown in Figure 1. Note that the shape of the spray generation function does not change as the wind speed increases, indicating that the relative contributions of film, jet and spume droplets are constant over this range of wind speeds.

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Figure 1. Variation of the Andreas sea spray generation function with the radius at formation for three wind speeds at salinity S = 30‰.

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2.2. Sea spray concentration measurements

Sea spray droplet concentrations near the ocean surface have been reported by many researchers. Some of the available spray concentrations, measured between 0.15 and 14 m above the mean surface, are summarized in Table I in order of increasing U10, from 0 to 28.8 m s−1. The droplet radii for the reported concentrations are as measured, not corrected to r0 or r80. For each case the location and date are given, along with available information on wind speed, relative humidity RH, water depth D, method used to measure concentration, and the range of droplet radii that were sampled by that method. The value for U10 is calculated from the measured wind speed at the specified height using Andreas's bulk turbulent flux algorithm (Andreas et al., 2008; Andreas, 2010).

Table 1. Sea spray concentration measurements. The column labeled U (hgt m) gives the measured wind speed followed by the anemometer height in metres. RH is relative humidity, U10 is the wind speed at 10 m, and D is the water depth. These parameters are not available for all cases.
Author JournalHgtUU10RH %DLocationMethodDroplet
 Year (hgt m)     radius range
 Figuremm s−1m s−1 mDate μm
Smith et al.QJRMS140 (14)077 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
       1986  
Monahan et al.QJRMS14 3various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
Smith et al.QJRMS145 (14)4.977 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
       1986  
Monahan et al.QJRMS14 5various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
de Leeuw 5     summer 1978  
 Tellus0.25.5 (25)5.270 N. Atlantic 6/4/1983wave-following Rotorod5 to 50
 1986        
 2        
de Leeuw Tellus0.27 (25)6.668 N. Atlantic 6/22/1983wave-following Rotorod5 to 50
 1986        
 3        
Monahan et al.QJRMS14 7various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
de Leeuw Tellus0.57.7 (10)7.78518North Seawave-following Rotorod>5
 1990     11/15/1986  
 5        
de Leeuw Aer Sci0.28.5 (25)7.964 N. Atlanticwave-following Rotorod>5
 1986     6/8/1983  
 2        
de Leeuw JGR0.58.9 (10)8.97518North Seawave-following Rotorod>5
 1987     11/20/1984  
 1        
Monahan et al.QJRMS14 9various  ASASP-300 CSASP-1000.1 to 15
Preobrazhenskii 1983     N. Atlantic near Rockall summer 1978  
 5        
 FM-SR1.5 to 27 to12 (7)7.2 to 12.4  N. Atlantic10 and 20 mm glass slides>5
 1973     fall 1969  
 2        
Smith et al.QJRMS1410 (14)9.777 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
       1986  
Monahan et al.QJRMS14 11various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
Monahan JGR0.1311 (10)11  Arubashadowgraph photography>45
 1968     February 1964  
 2        
de Leeuw Tellus0.213 (25)11.968 N. Atlantic 6/13/1983wave-following Rotorod5 to 50
 1986        
 4        
de Leeuw Tellus0.512 (10)128018North Seawave-following Rotorod>5
 1990     10/22/1986  
de Leeuw 6        
 JGR0.512.4 (10)12.48018North Seawave-following Rotorod>5
 1987     11/21/1984  
 2        
Monahan et al.QJRMS14 13various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
de Leeuw Tellus0.514 (10)147518North Seawave-following Rotorod>5
 1990     10/24/1986  
 7        
Monahan et al.QJRMS14 15various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
Taylor and Wu JGR8.214.9 (8.2)15.2669.1585-m-longlaser and photodiode25 to 250
 1992     pier in NC  
 2     spring 1983  
Monahan JGR0.1316 (10)16 11Buzzards Bay 5/22/1964shadowgraph photography>45
 1968        
 3        
Taylor and Wu JGR8.216 (8.2)16.4679.1585-m-long pier inlaser and photodiode25 to 250
 1992 2     NC spring 1983  
Monahan et al.QJRMS14 17various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
Preobrazhenskii FM-SR1.5 to 215 to 25 (7)15.6 to 26.2  N. Atlantic10 and 20 mm glass slides>5
 1973     fall 1969  
 2        
Monahan et al.QJRMS1418+ (10)18+various N. AtlanticASASP-300 CSASP-1000.1 to 15
 1983     near Rockall  
 5     summer 1978  
Smith et al.QJRMS1420 (14)19.377 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
       1986  
Smith et al.QJRMS1425 (14)2477 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
de Leeuw       1986  
 Tellus1125 (10)256518North Seawave-following Rotorod>5
 1990     11/15/1986  
 4        
Smith et al.QJRMS1430 (14)28.877 N. AtlanticFSSP0.1 to 23.5
 1993     at island ofASAP-300 
 1     South Uist  
       1986  

There is important information about the concentration data that cannot be summarized in the table, as well as exceptions to the norm in reporting concentration measurements:

  • Monahan et al. (1983) made measurements at various relative humidities and adjusted the measured radii and concentrations to values at equilibrium at RH = 80%. They grouped their concentration data in 2 m s−1 increments in U10, using the average wind speed for the 6 h immediately before each sampling interval of 1200 s.

  • Monahan (1968) compiled data from 5 days during a 4 week period in Aruba into a single concentration distribution.

  • Taylor and Wu (1992) adjusted the droplet radii they measured to radii at equilibrium with the relative humidity at the sea surface for a salinity of 28.5‰. They averaged concentration data over periods of monotonically increasing or decreasing wind speed.

To compare these measured concentration distributions with each other and with the spray generation function, we converted concentrations that were reported in terms of droplet diameter d to droplet radius r:

  • equation image(1)

2.3. Equilibrium assumption

What is the relationship between spray concentration dC/dr and spray generation dF/dr? According to Fairall et al. (2009) when the concentration profile is near equilibrium, the spray source strength is given by the spray concentration near the ocean surface multiplied by the gravitational fall velocity vg of the droplets. For droplet radii between 5 and 50 µm, vg ranges from about 0.003 to 0.3 m s−1. It reaches 1 m s−1 for a droplet radius of 125 µm and 2 m s−1 at 250 µm.

The 31 cases from Table I are presented in 12 plots in Figure 2 in order of increasing U10 from 0 to 28.8 m s−1. We use the concentration data as presented by the authors without adjusting the ambient droplet size r to either r0 or r80. Note also that we are using the Andreas spray generation function slightly outside the 5 m s−1< U10 < 25 m s−1 range for which it is intended. The horizontal axis in each plot is the logarithm of droplet radius, ranging from 4 to 400 µm. The vertical axis is dF/dr0 (m−2s−1 µm−1) and dC/dr (m−3 µm−1) on the same seven-decade log scale. By plotting the measured concentrations and the calculated generations on the same logarithmic scale, we show graphically the vertical droplet velocity that is required for the spray generation function to be consistent with the measured spray concentrations. For example, in Figure 2(d), droplets of 20 and 40 µm radius need vertical velocities of about 1 m s−1 and 100 m s−1, respectively, to exhibit the droplet flux specified by the spray generation function.

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Figure 2. Points: measured dC/dr (m−3 µm−1) in order of increasing U10. Lines: calculated dF/dr0 (m−2 s−1 µm−1) for the same U10. When two lines are shown, they represent dF/dr0 for the lowest and highest wind speeds in the measurements. The vertical distance between the concentration points and generation lines represents the production velocity (m s−1) for that droplet radius that is required for them to be compatible.

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Concentration measurements at similar wind speeds in Figure 2 are grouped in one plot, along with the calculated spray generation function for the wind speed or the range of wind speeds for those measurements. For the Preobrazhenskii cases in Figure 2(f) and (j), for example, the two spray generation curves are for the range of wind speeds that he reported. That range brackets the curves for the wind speeds associated with the other concentration data in those plots.

Some of our plots of the concentration data differ from the referenced papers; specifically:

  • The two Preobrazhenskii (1973) cases appear to have no correction for the collision efficiency of the droplets with his glass slides. Therefore, we corrected the reported concentrations for collision efficiency. This correction is only approximate because (i) glass slides 10 and 20 mm wide were used and collision efficiency decreases as width increases, and (ii) he averaged concentrations for a wide range of wind speeds and collision efficiency increases with wind speed. We also merged his droplet bins into bins with a 5 µm range in radius

  • For the Smith et al. (1993) and the Taylor and Wu (1992) cases, we show about half their data to indicate the trend of the concentration data.

  • For the de Leeuw (1986a,b, 1987, 1990) concentration profile data, the lowest height for the concentration profile is reported in Table I. However, if the concentration increased within 1 m above that height, we use that higher concentration in Figure 2.

  • For the two Taylor and Wu (1992) cases, we provide in Figures 2(h) and (i) both the data they report (open circles) and the concentrations for the ambient droplet radii (solid circles), assuming, following Lewis and Schwartz (2004), that the ambient radii at RH = 65% were all adjusted to radii in equilibrium with seawater at S = 28.5‰.

With the exception of the Monahan (1968) measurements, the concentration data in Figure 2 are remarkably consistent, particularly in light of the varying elevations at which the measurements were made, the different methods that were used to measure droplet concentration and the various ways the associated wind speed was specified.

Also interesting about Figure 2 is the lack of droplets with ambient radii greater than about 50 µm for U10 < 16 m s−1 in the concentration measurements. Some of these concentration measurements were made by de Leeuw and Preobrazhenskii, both of whom measured concentration distributions with radii larger than 50 µm for larger U10. Specifically, Preobrazhenskii reported concentrations for ambient radii up to about 200 µm for U10 between 15.6 and 22.2 m s−1 (Figure 2(j)), and de Leeuw reported concentrations for ambient radii up to 115 µm for U10 = 25 m s−1 (Figure 2(k)). The only case with radii larger than 50 µm for U10 < 16 m s−1 is Monahan (1968) for U10 = 11 m s−1 (Figure 2(f), downward pointing triangles) for measurements at 0.13 m above the ocean surface. Those near-surface droplets may be short-lived, never making it above wave height, in which case they would not be part of the effective spray generation. In contrast, the spray generation function produces droplets with r0 between 100 and 400 µm and larger at significant rates for U10 < 16 m s−1.

If the source strength and concentration of droplets are related by the droplet gravitational fall velocity as Fairall et al. (2009) suggest, the concentration data in Figure 2 would be about two decades above the spray generation curve at 10 µm (vg = 0.01 m s−1), coincide with the curve at 125 µm (vg = 1 m s−1) and be about a half decade below the curve at 400 µm (vg = 3.3 m s−1). The plots in Figure 2 do not support this relationship.

To show explicitly the difference between the inferred spray generation distribution and the Andreas spray-generation function, we plot in Figure 3vgdC/dr calculated for three of the Figure 2 wind-speed increments (c, g and k), representing moderate, high and very high winds. We show for comparison both dF/dr0 and dF/dr80 because the droplets sampled in the concentration measurements will have lost mass to evaporation, with some of them probably reaching their equilibrium size at the local humidity, and thus may be better represented by dF/dr80. Figure 3 shows that the inferred spray generation distribution is smaller than the spray generation function by about an order of magnitude. Furthermore, comparing Figures 3(a) and (b), for moderate and high wind speeds, with Figure 3(c), for very high wind speeds, we see a change in shape of the inferred spray generation distribution.

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Figure 3. Spray generation function (solid line dF/dr0 plotted against r0, dashed line dF/dr80 plotted against r80) and inferred spray generation calculated from measured concentrations and droplet fall velocity: (a) moderate winds, (b) high winds and (c) very high winds. The radius on the horizontal axis is either the ambient radius or r0 or r80.

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This lack of agreement between the measured spray concentration distribution and the spray generation function has a few possible explanations:

  • None of the measured concentrations happened to be at equilibrium with spray generation for the reported wind speed. However, if equilibrium is generally satisfied, it is unlikely that it would not be satisfied in all these cases.

  • The concentration equilibrium assumption is not satisfied. A number of authors have used velocities other than vg to relate concentration and generation. Monahan (1968) calculated the production rate of droplets 150 µm in radius as the product of the measured concentration and the ejection velocity of jet droplets vj. Jones and Andreas (2009) noted the generation–concentration discrepancy and found that spray concentrations based on either vj or the friction velocity u provided much better agreement with icing observations on offshore platforms. Andreas et al. (2010) examined a number of possible spray droplet production velocities and found that the mean wind speed at wave crest height and vj have the right magnitude over the range 20 µm < r0 < 300 µm.

  • The Andreas spray generation function gives sea spray fluxes that are too large. Fairall (2010, personal communication) has decided in the past few years that the Andreas/Fairall source function is on the high side.

In order to simulate icing on offshore platforms we need a spray concentration function. By comparing simulated and observed icing rates we can test concentration functions. Functions that provide a reasonable match to the observations can then be used to estimate the spray generation function.

3. Spray concentration function

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

Lewis and Schwartz (2004, pp. 17, 174) provide a canonical concentration function dC (r80)/dlnr80, which we rewrite in terms of dC(r80)/dr80 [m−3 µm−1]:

  • equation image(2)

The parameters here are the geometric mean r80 = 0.3 µm, the geometric standard deviation σ = 2.8 and maximum amplitude n0 = 7 × 104 U102, with U10 in m s−1. This empirical function, which has a factor of three uncertainty, holds for U10 between 5 and 20 m s−1 and for r80 between 0.1 and 25 µm. According to Lewis and Schwarz (2004, p. 166), the upper radius limit represents the largest droplets that are significant in sea spray. We have plotted Eq. (2) in Figure 4(a) for U10 = 6 and 12 m s−1, extending the curves to a minimum concentration of 0.2 m−3 µm−1. The measured concentrations for the ambient droplet radii at similar wind speeds from Figure 2(c) and (g), also shown in Figure 4(a), agree well with Eq. (2). Note that there were measurable concentrations for ambient droplet radii extending to about twice the r80 = 25 µm Lewis and Schwartz maximum.

thumbnail image

Figure 4. (a) Equation (2), film- and jet droplet concentration function (lines) for U10 = 6 and 12 m s−1 compared with measured concentrations from Figures 2(c) and (g); (b) Equation (3), spume droplet concentration function (lines) for U10 = 20, 24 and 28 m s−1 compared with measured concentrations from Figures 2(j), (k) and (l).

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Spume droplets do not contribute to Eq. (2) (Lewis and Schwartz, 2004, p. 175). However, Monahan et al. (1983) state that their data show a pronounced enhancement of large droplets for U10 between 9 and 11 m s−1, indicating the onset of spume droplet production. Their spectra from U10 = 3 m s−1 to U10 = 18+ m s−1 are shown in Figure 5, transformed from volume distributions to concentration distributions. The difference between the shapes of the U10 = 3 to 9 m s−1 and the U10 = 11 to 18+ m s−1 curves is slight. Furthermore, this difference does not require the onset of spume droplet production; it may be simply an enhancement in the turbulent transport of the larger droplets to the 14 m sampling elevation at higher wind speeds. We follow Lewis and Schwarz (2004) in attributing concentration distributions for U10 less than about 20 m s−1 to film and jet droplets generated by the bursting of bubbles in whitecaps. We assume that spume production becomes significant at higher U10.

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Figure 5. Concentration data from Monahan et al. (1983) measured at a 14 m elevation. The legend indicates 2 m s−1 increments in average U10, from J1 = 3 m s−1 to J9 = 18+ m s−1.

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In Figure 4(b) we plot measured dC/dr for U10 = 19.3, 24, 25 and 28.8 m s−1 from Figure 2(j), (k) and (l). At these wind speeds, droplets with ambient radii as large as about 100 µm were observed at measurable concentrations at heights of 11 and 14 m above the sea surface. The measured concentrations at these higher wind speeds are not consistent with Eq. (2), either in the relative abundance of large droplets or in the rapid increase in concentration with U10. Therefore, we varied σ and n0 from the values in Eq. (2) in order to obtain a reasonable eyeball match to the data. The concentration function that includes spume droplets with σ = 4 and n0 = 30 U104 is

  • equation image(3)

which is plotted in Figure 4(b) for U10 = 20, 24 and 28 m s−1. The power of U10 in n0 is not well determined in Eq. (3) because of the small range of very high wind speeds in this sample. It is at least four, but may be greater.

4. Spray droplet concentration profiles

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

We have plotted concentrations that were measured anywhere from 0.13 m to 14 m above the sea surface in Figures 2–4, implying that they are comparable. Is that correct? Fairall et al. (2009) provide a formulation for the decrease in spray concentration with height:

  • equation image(4)

with Kármán constant κ = 0.4 and slip factor fs ≈ 1. The height h defines the upper limit of the source region for spray droplet generation. Equation (2) gives dC(r,h)/dr for U10 < 19 m s−1 and Eq. (3) is used for greater wind speeds. We can compare concentration ratios from the profiles published by de Leeuw (1986a,b, 1987, 1990) with the power law decrease with height in Eq. (4).

The concentration ratios shown in Figure 6 are the measured concentration at the top of the profile divided by the maximum concentration measured between 0.2 and 2 m above the surface of the ocean. This concentration ratio characterizes the overall decrease in concentration with height. The curves in each plot are calculated from Eq. (4), using a nominal h = 1 m because film and jet droplets are produced from bubbles in whitecaps, generally in the wave troughs on the windward side of the crest (de Leeuw, 1990). Typical maximum profile heights are z = 11 and 21 m, but may be as low as 6 m when concentrations at higher levels for a particular droplet size were not measurable. Those lower level ratios are indicated in the plots in Figure 6. Note that for the U10 = 11.9 m s−1 case, the concentration ratios are all relatively low because of high concentrations near the ocean surface remaining from earlier stormy conditions, rather than because of low concentrations aloft (see the high concentrations for U10 = 11.9 m s−1 compared with 12.4 m s−1 in Figure 2(g)). In many of the other plots in Figure 6, the measured concentration ratios exceed the power law ratio for some droplet sizes. In some cases the concentration aloft is greater than that near the ocean surface. For the most part, concentrations at some distance above the ocean surface are within a factor of three of the concentration near the droplet source and thus are within the uncertainty in Eq. (2).

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Figure 6. Spray concentration ratios calculated from Equation (4) (z = 11 m solid line and z = 21 m dashed line), with h = 1 m, compared with ratios for z = 11 and 21 m from the de Leeuw profiles (1986a,b, 1987, 1990). Ratios for maximum profile heights less than 11 m are annotated with the height of the profile.

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There are no measured concentration profiles for the very high wind speed cases in Table I. Spume droplets are generated by the wind tearing water off the crests of waves. Therefore, to determine concentration profiles using Eq. (4) from the spume concentration function Eq. (3), we specify h = 0.5 H1/3.

The liquid water content of the spray is

  • equation image(5)

where ρw is the density of water. The functions dW/dr and dC/dr are compared in Figure 7 at h, 10 m and 30 m for U10 = 12 and 24 m s−1. At 12 m s−1, the droplets are generated by bursting bubbles so h = 1 m. At U10 = 24 m s−1, which is above the spume threshold, h = 0.5H1/3.

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Figure 7. (a) Droplet concentration and (b) liquid water content for sea spray at height h (top curve in each triad) and at 10 and 30 m (middle and bottom curves), for U10 in the film and jet droplet range (12 m s−1, solid lines) and in the spume droplet range (24 m s−1, dashed lines).

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In the next section we use this profile of liquid water in the spray to simulate sea spray icing on fixed offshore platforms. Comparing our modelled icing rates with observations of ice accretion provides a check of Eq. (2), the Lewis and Schwartz (2004) spray droplet concentration for film and jet droplets for moderate and high winds, and Eq. (3) for the very high wind speeds that generate spume droplets.

5. Ice accretion on fixed offshore structures

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

We have weather data and icing information for two offshore structures; for a single event on the Sedco 708 platform and for a winter season on the Ocean Bounty platform. In this section we compare observations of ice accretion with simulated ice thicknesses and icing rates based on the sea spray concentration functions, Eqs (2) and (3), the spray droplet profile Eq. (4) and the spray water content Eq. (5). For the ice accretion simulations we use measured wind speeds and significant wave heights and take into account the collision efficiency of the spray droplets with structural components.

We calculate U10 and u from the wind speed measured at a specified height using Andreas's bulk turbulent flux algorithm (Andreas et al., 2008; Andreas, 2010). We then calculate dC/dr near the ocean surface using Eq. (2) for U10 < 19 m s−1 and using Eq. (3) for higher winds. With u and vg, we determine dC(r,z)/dr from Eq. (4), with h = 1 m for U10 < 19 m s−1 and h = 0.5H1/3 for higher wind speeds.

The spray droplets quickly cool to a temperature below the air temperature T. Therefore, if T (z) < 0°C, we assume that the sea spray freezes to components on the platform at height z. From the spray liquid water content dW(r,z)/dr in Eq. (5) we determine the icing rate on cylinders with axes perpendicular to the wind direction:

  • equation image(6)

Here, I(z) is ice thickness on the front of a cylinder with diameter D, t is time, U(z) is wind speed at height z, ρi is ice density and E(U, r, D) is the collision efficiency of the droplets with the cylinder. We take rmin = 5 µm and rmax = 100 µm for U10 < 19 m s−1 and 200 µm otherwise. The collision efficiency, between 0 and 1, is the fraction of the droplets swept out by the cylinder that actually collide with it. Collision efficiency results from a balance between inertia, which tends to keep the droplets travelling in a straight line, and drag, which tends to make the droplets follow the wind streamlines around the cylinder. Collision efficiencies for cylinders were computed and provided in graphical form by Langmuir and Blodgett (1946) and recomputed with an improved drag formulation and provided as formulae by Finstad etal. (1988).

The focus in this article is on the flux of spray to the platform so we make simplifying assumptions about the formation of ice from the spray droplets: (i) the droplets freeze when they hit the platform if the air temperature is less than 0°C; (ii) all of the impinging water freezes on impact; (iii) the density of the saline ice is the same as the density of glaze ice, ρi = 900 kg m−3; and (iv) the ice freezes on the front of the cylinder (representing a structural component or equipment) so that its diameter does not increase and E does not decrease as ice accumulates. In reality: (i) the freezing temperature depends on the salinity of the water; (ii) the water may flow around the cylinder or drip off before freezing; (iii) the density of ice formed from sea spray depends on the amount of air and water incorporated in the ice matrix (Ryerson and Gow, 2000); and (iv) the diameter of an ice-covered cylinder is likely to increase as more ice accumulates on it, decreasing the collision efficiency as spray icing continues. All of these complications can be taken into account in a complete sea spray icing model.

5.1. Sedco 708

The semi-submersible drilling rig Sedco 708 was deployed by ARCO Alaska for the Continental Offshore Stratigraphic Test on the North Aleutian Shelf (56.274°N, 161.976°W) in 53 m of water. The Sedco 708 main deck is about 39 m above the bottom of the pontoons, so at the normal drilling draft of 26 m, the main deck is about 13 m above the ocean surface. An ice accretion observation programme (Minsk, 1984a, 1985) was undertaken alongside the drilling programme. For the icing programme, six arrays of cylinders were mounted just below the main deck, where they had reasonable exposure to sea spray icing.

Only two significant storms occurred while the drilling rig was on station: one in early December 1982 that was not documented, and one from 3 to 8 January 1983. During the January event, as the air temperature (Figure 8(a)) decreased to below −10°C and the water temperature (not shown) decreased from 3.9 to 3.3°C, U10 (Figure 8(b)) increased to just over 19 m s−1. Measured significant wave heights (Figure 8(c)) reached almost 6 m and agree well with H1/3 values calculated with the Andreas and Wang (2007) algorithm.

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Figure 8. January 1983 spray icing on the Sedco 708: (a) air temperature, (b) U10, with the 19 m s−1 spume threshold shown with a dashed line, (c) observed and calculated H1/3 and (d) calculated accumulated spray ice thickness at 5 m.

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No icing was observed on the derrick above the main deck or on the railing of the main deck. There was very little ice accumulation on the cylindrical trusses below the main deck on the windward side of the platform. Waves kept the lower third of the trusses clear of ice. Above that, however, up to about 5 inches (127 mm) of ice accumulated on the trusses in the middle of the below-deck structure. From there the ice thickness decreased towards the bottom of the main deck. The maximum ice thickness on the cylinder arrays mounted just below the main deck around the perimeter of the structure was estimated from photographs to be about 1 inch (25 mm). The array with the maximum thickness was on the lee side of the platform.

The icing rate for each hour at heights from 5 to 50 m above the ocean surface was calculated from U10 and h (either 1 m or H1/3), assuming a 100% collision efficiency to obtain the maximum possible ice thickness. The accumulation of ice at 5 m, which is just above the lower third of the trusses, is shown in Figure 8(d). The 3 mm accumulation is consistent with the observed lack of ice on the windward side of the platform structure at this level. It is likely that the ice observed in the middle of the below-decks structure and on the lee-side cylinder array was the accumulation of droplets that were created by the splashing of the waves against the trusses and subsequently carried downwind underneath the main deck.

5.2. Ocean Bounty

Spray icing of the semi-submersible exploratory drilling rig Ocean Bounty during the winter of 1979–1980 is described in Williams (1981), Nauman (1984), Nauman and Tyagi (1985) and Minsk (1984b). The Ocean Bounty is 107 m long and 81 m wide, with the main deck 16 m above the ocean surface. The rig was operated by Phillips Petroleum Company near Kamishak Bay in Lower Cook Inlet, Alaska, 20 km from shore in 160 m of water. In this region wind speeds can vary dramatically over short distances because of funnelling and channelling of the air through gaps in the nearby mountains.

From 24 September 1979 to 26 April 1980, 21 days of sea spray icing on the superstructure (Figure 9) in six storms lasting between 1 and 7 days each were documented. Meteorological and oceanographic data were recorded by Oceanroutes, Inc., meteorologist-observers every day. Beginning 20 December 1979, superstructure icing was added to the daily data sheets and was recorded as light, moderate, heavy, or very heavy, following Wise and Comiskey (1980). They quantify icing rates in inches per day as 0.04 to 1.4 (light), 1.4 to 2.6 (moderate), 2.6 to 5.7 (heavy), and 5.7+ (very heavy). There is no information about where on the OceanBounty the icing observations were made, how the icing rate was estimated, and the orientation of the ice-covered surface relative to vertical. Weather, sea and icing parameters were recorded every 2 h from 0600 to 1800 LST. Some of those parameters were also recorded at 0200 and 2200. Water temperature was measured once a day at 1400. The anemometer was 84 m above sea level.

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Figure 9. Sea spray icing on the Ocean Bounty in the winter of 1979–1980 in Lower Cook Inlet (photograph provided by Minerals Management Service).

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Time series of air and water temperature, U10, and observed and calculated H1/3 are shown in Figure 10 for the 87 days in the winter of 1979–1980 with air temperatures near or below 0°C for at least a portion of the day. Wind speeds were above the 19 m s−1 spume threshold for extended periods. The observed H1/3 is much less than that calculated from U10 with the Andreas and Wang (2007) algorithm, probably because of the limited fetch, with H1/3(observed) = 0.32 H1/3(calculated) on average. When H1/3 was not recorded, we used this relationship to estimate it from the calculated value.

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Figure 10. Ocean Bounty weather conditions and spray icing: (a) measured air and water temperatures, (b) U10 with the 19 m s−1 spume threshold shown by a dashed line, (c) observed and calculated significant wave heights and (d) icing rate observations and the calculated icing rate at 20 m on a 5 cm diameter cylinder.

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The simulated icing rate at an elevation of 20 m on a cylinder 5 cm in diameter is shown in Figure 10(d) along with the observed estimates of superstructure icing severity, plotted at the midpoint of the Wise and Comiskey (1980) levels and at 8 inches per day (≈8 mm h−1) for ‘very heavy’ icing. We do not know the elevation and size of the component on which the icing observations were made. If it was lower than 20 m or if the diameter was less than 5 cm, the simulated icing rate would be larger than shown in Figure 10(d).

It appears from Figure 10 that the recording of ice accumulation began with the second significant spray icing event, the meteorologists perhaps realizing then that spray icing was not going to be a rare occurrence. The timings of the subsequent modelled and observed spray icing storms coincide, with the observed and calculated icing rates agreeing well for three of the six events. For the other three, the observed icing rates are higher than the calculated rates. Two of those events are characterized by very cold temperatures and somewhat lower wind speeds that would probably result in lower density ice with more air inclusions. The mass accretion rates of this thicker, low-density ice may, therefore, be consistent with our modelled rates.

The median volume droplet radius rMVR is useful for characterizing the droplet size in a cloud or in sea spray. It is the radius for which half the liquid water is in smaller droplets and half is in larger droplets. In sea spray rMVR would be expected to increase with wind speed and decrease with height above the ocean surface. For wind speeds less than 19 m s−1 (average U10 = 12.6 m s−1) during the Ocean Bounty icing episodes, rMVR was relatively small and decreased only slightly from 9.5 to 9.3 to 9 µm with height increasing from 10 to 20 to 40 m. For higher wind speeds (average U10 = 25.8 m s−1), rMVR was much larger and decreased significantly for this increase in height: from 37 to 29 to 24 µm. This shows the importance of the large droplet range of the concentration function Eq. (3). As an additional check of the significance of Eq. (3), we calculated the accumulation of ice on the Ocean Bounty using the concentration function Eq. (2) for all wind speeds. That resulted in virtually no simulated ice accumulation, indicating that our spume formulation Eq. (3) is necessary to represent spray generation at these very high wind speeds.

5.3. Implications for spray generation

The spray generation function that is consistent with the concentration functions, Eqs (2) and (3) and the equilibrium assumption is

  • equation image(7)

That derived spray generation function is shown in Figure 11 for U10 = 6, 12, and 24 m s−1. Winds of 24 m s−1 generate about ten times as many spray droplets with r0 > 4 µm as 12 m s−1 winds, which in turn generate droplets at four times the rate of 6 m s−1 winds. While the generation rate increases with wind speed, the distribution of droplet radii is the same for winds up to 19 m s−1, with a median volume radius at formation of about 55 µm. Spume droplets, which are generated in addition to film and jet droplets for U10 > 19 m s−1, increase rMVR at the source to about 220 µm. For comparison, Andreas (2002) estimates practical maximum r0s for jet and spume droplets at 100 and 500 µm, respectively. The generation of spume droplets by very high winds led to the high icing rates that were observed on the OceanBounty.

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Figure 11. Variation with wind speed of the sea spray generation function derived from the concentration functions (Eqs (2) and (3)) and the equilibrium assumption.

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6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

By directly comparing a spray generation function with measured droplet concentrations, we showed that the concentration equilibrium assumption (e.g., Fairall et al., 2009) is not satisfied. The vertical velocities of the spray droplets would have to be at least an order of magnitude larger than the gravitational fall velocity for the product of the concentration and the droplet velocity to equal the Andreas spray generation function. This lack of agreement may be because the Andreas/Fairall generation function is too high, as Fairall (2010, personal communication) suggests. If, on the other hand, this generation function is correct, then the lack of agreement requires specifying a production velocity different from the gravitational fall velocity, as was done recently by Jones and Andreas (2009) and Andreas et al. (2010).

Because of the lack of concurrence between the spray generation function and measured concentrations, we calculated spray droplet profiles starting with the Lewis and Schwartz (2004) droplet concentration function Eq. (2). That function agrees well with measured concentrations at moderate and high wind speeds. We extended it to very high wind speeds, U10 > 19 m s−1, to include spume droplets, increasing σ from 2.8 to 4 and changing the wind speed dependence from n0 = 7 × 104 U102 to n0 = 30 U104. The parameters of the droplet concentration function Eq. (3) for very high wind speeds are uncertain because of the limited datasets available for defining them. It is clear, however, that the tail of that function is longer than the tail of the concentration function for film and jet droplets and that n0 has a stronger wind speed dependence. Future work should focus on better defining those parameters and their relation to the physics of spume generation.

We used the Fairall et al. (2009) power law decrease in droplet concentration above the spray source region. It agrees reasonably well with concentration ratios from the de Leeuw profiles. In this comparison we used the source height h = 1 m because the whitecaps where the droplets are generated occur between the wave crests.

We applied this formulation for spray concentration profiles to spray icing of fixed offshore platforms at moderate to very high wind speeds. While ships create spray by slamming into waves as they push through the wave field, spray on fixed platforms with a small waterline area comes primarily from droplets generated from wind waves. The very small accumulation of ice—both observed and modelled—on the windward side of the Sedco 708 platform in an icing episode 5 days long shows that there is little liquid water in the spray cloud generated by wind speeds up to the spume threshold. On the other hand, the winds experienced by the Ocean Bounty, which often greatly exceeded 19 m s−1, resulted in dramatic ice accumulations. At these wind speeds, large spume droplets are ripped off the crests of high waves. The droplets are readily carried aloft, and their inertia tends to make them collide with components on the superstructure rather than following the wind streamlines. Our simulation showed good agreement with the rough observations of the rate of ice accumulation on both the Sedco 708 and the OceanBounty.

The spray generation function derived from the concentration functions and the equilibrium assumption has a median volume radius at generation in the jet droplet range for U10 < 19 m s−1 and in the spume droplet range for higher wind speeds.

With the observed retreat of the Arctic sea ice cover and an increase in oil and natural gas exploration and production in northern seas, the frequency of icing on offshore platforms will increase. These platforms could be exploited to measure sea spray concentrations in conditions not encountered in temperate waters. Instituting a more rigorous icing observation programme than was done on the Sedco 708 or Ocean Bounty, including measurements of ice accretion on cylinders at various heights in the superstructure, would provide data on the profiles of liquid water and droplet radius in sea spray. With measurements made over a wide range of wind speeds we could test our proposed spray droplet concentration function for very high wind speeds.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References

This work was supported by US Department of the Interior Minerals Management Service (MMS) under Interagency Agreement M07RG13274. The US Office of Naval Research also supported ELA through grant N00014-08-1-0411. We thank Kyle Monkelien of MMS for providing the original Ocean Bounty ballast reports for the winter of 1979 to 1980, Chet Miller of MMS for providing photographs of the Ocean Bounty, and Chip Alvord for providing information about the layout of Sedco 708.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea spray generation and concentration distributions
  5. 3. Spray concentration function
  6. 4. Spray droplet concentration profiles
  7. 5. Ice accretion on fixed offshore structures
  8. 6. Conclusions
  9. Acknowledgements
  10. References
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